MathDB

2

Part of 2001 Turkey Team Selection Test

Problems(2)

Coaxal Circles

Source: Turkey TST 2001 - P2

4/4/2013
A circle touches to diameter ABAB of a unit circle with center OO at TT where OT>1OT>1. These circles intersect at two different points CC and DD. The circle through OO, DD, and CC meet the line ABAB at PP different from OO. Show that ∣PA∣⋅∣PB∣=∣PT∣2∣OT∣2.|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.
geometrygeometric transformationreflectiongeometry proposed
The line through orthocenter and midpoint

Source: Turkey TST 2001 - P5

4/4/2013
Let HH be the intersection of the altitudes of an acute triangle ABCABC and DD be the midpoint of [AC][AC]. Show that DHDH passes through one of the intersection point of the circumcircle of ABCABC and the circle with diameter [BH][BH].
geometrycircumcirclegeometry proposed